Hi Mahdi, J is the torsion constant, this is related to the second moment of area : I (also known as; area moment of inertia, polar moment of area/inertia or second area moment) in the case of a circle or concentric circular hollow pipe, the constant has to be worked out for more complex cross sections. Implementing FEA for more complex problems is advantageous because of the convolution of stress concentrations in non-unifrom shapes.
Yes, these are the classical ways to denote torsional stiffness and bending stiffness. However, for complex cases, the use of intertia tensors is necessary.
You may check the following books for some additional details:
* Handbook of continuum mechanics. General concepts, thermoelasticity,
Jean Salençon, Springer.
* Elasticity and Geometry: From hair curls to the nonlinear response of shells by Basile Audoly and Yves Pomeau, Oxford University Press
Mahdi
$J_{rr}$ is the polar moment of inertia, it is evaluated as follows: $J_{rr} = \int_0^R r^2 dA$. So for a filled cylinder of outer radius R, this polar moment is equal to: J_rr = pi/2*R^4.
I_{ij} is the area moment of inertia Tensor so $I_{ij} = 0.5*\int_A (x_i^2 + x_j^2) dA$.
Thus for a circular section: I_rr = I_xx + I_yy.
This last formula is true for all cross-section that has an axis of symmetry in the axial direction
J is the torsional rigidity factor that is equal to the polar inertia only when the beam cross section is circular (i.e., without warping in torsion). Otherwise, if you consider a general cross section, the quantity J is smaller than the polar inertia due to warping effects. Please, refer to standard textbkoks of structural mechanics.
'G' is the torsional Rigidity and 'I' is Moment of Inertia of the cross section.
Both are area moment of Inertia or 2nd moment of area, but both are about different axes. J being about an axis normal to the plane and I being about an axis parallel to plane. Area moment of inertia is basically represents the resistance (sluggishness) to bending or twisting just like mass (or mass inertia) represents the resistance to motion.
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